The Discrete Cosine Transform (DCT) is an irrational transform commonly used in lossy image compression. The DCT is commonly used in lossy image compression. It is used in many modes of the JPEG standard and the MPEG standards and future HDTV in the United States. For a discussion on the various standards, see ISO Standard documents ISO/IEC 10918 (JPEG), 11172 (MPEG 1), 13818 (MPEG 2) and William B. Pennebaker and Joan L. Mitchell, “JPEG Still Image Data Compression Standard,” 1993. The basis vectors of DCT have irrational values. Theoretically, integer inputs result in irrational transform coefficients. Therefore, infinite precision is required to perform those transforms exactly. For use in compression, transform coefficients must be rounded to a finite representation.
With most transform implementations, the rounding coefficients to integers does not guarantee that every unique integer input results in a different output. Therefore, the inverse DCT cannot reconstruct the input exactly. The error due to forward and inverse DCT transforms without quantization is referred to as systemic error. This systemic error prevents DCT implementations from being used in lossless compression without retaining a difference or error image.
In practical DCT implementations, the transform basis vectors are also rounded. The difference between a given implementation and the ideal transform (or a high accuracy floating point implementation) is referred to as mismatch. Low mismatch is required for data interchange. There can be a trade-off between the amount of mismatch and speed, cost and other desirable features.
A parameterized transform referred to herein as the Allen Parameterized Transform (APT) is a family of fast transforms which can implement the DCT or rational transforms that are arbitrarily close to the DCT. The APT is also referred to as a generalized Chen transform (GCT) For more information on the APT, see J. Allen, “Generalized Chen Transform: A Fast Transform for Image Compression,” Journal of Electronic Imaging, Vol. 3(4), October 1994, pgs. 341-347; J. Allen, “An Approach to Fast Transform Coding in Software,” Signal Processing: Image Communication, Vol. 8, pp. 3-11, 1996; and U.S. Pat. No. 5,129,015.
The present invention provides a reversible block based transform, such as, for example, a reversible DCT. The DCT of the present invention may be included in a DCT-based compressor/decompressor that may be used in a lossless compression/decompression system. The present invention also provides DCT transforms with no systemic error and no (or low) mismatch.